Discretization error cancellation in electronic structure calculation: toward a quantitative study

Cancès, Eric; Dusson, Geneviève

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1617-1636.

It is often claimed that error cancellation plays an essential role in quantum chemistry and first-principle simulation for condensed matter physics and materials science. Indeed, while the energy of a large, or even medium-size, molecular system cannot be estimated numerically within chemical accuracy (typically 1 kcal/mol or 1 mHa), it is considered that the energy difference between two configurations of the same system can be computed in practice within the desired accuracy. The purpose of this paper is to initiate the quantitative study of discretization error cancellation. Discretization error is the error component due to the fact that the model used in the calculation (e.g. Kohn−Sham LDA) must be discretized in a finite basis set to be solved by a computer. We first report comprehensive numerical simulations performed with Abinit [X. Gonze, B. Amadon, P.-M. Anglade et al., Comput. Phys. Commun. 180 (2009) 2582–2615; X. Gonze, J.-M. Beuken, R. Caracas et al., Comput. Materials Sci. 25 (2002) 478–492] on two simple chemical systems, the hydrogen molecule on the one hand, and a system consisting of two oxygen atoms and four hydrogen atoms on the other hand. We observe that errors on energy differences are indeed significantly smaller than errors on energies, but that these two quantities asymptotically converge at the same rate when the energy cut-off goes to infinity. We then analyze a simple one-dimensional periodic Schrödinger equation with Dirac potentials, for which analytic solutions are available. This allows us to explain the discretization error cancellation phenomenon on this test case with quantitative mathematical arguments.

X. Blanc, É. Cancés and M.-S. Dupuy, Variational projector augmented-wave method. C. R. Math. Acad. Sci. Paris (2017). | MR

P.E. Blöchl, Projector augmented-wave method. Phys. Rev. B 50 (1994) 17953–17979. | DOI

E. Cancès, R. Chakir and Y. Maday, Numerical analysis of the planewave discretization of some orbital-free and Kohn−Sham models. ESAIM: M2AN 46 (2012) 341–388. | DOI | Numdam | MR | Zbl

E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralík, A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 352 (2014) 941–946. | DOI | MR | Zbl

E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralík, A perturbation-method-based post-processing for the planewave discretization of Kohn–Sham models. J. Comput. Phys. 307 (2016) 446–459. | DOI | MR | Zbl

E. Cancès, V. Ehrlacher and Y. Maday, Non-consistent approximations of self-adjoint eigenproblems: application to the supercell method. Numer. Math. 128 (2014) 663–706. | DOI | MR | Zbl

H. Chen, X. Dai, X. Gong, L. He and A. Zhou, Adaptive finite element approximations for Kohn–Sham models. Multisc. Model. Simul. 12 (2014) 1828–1869. | DOI | MR | Zbl

H. Chen, X. Gong, L. He, Z. Yang and A. Zhou, Numerical analysis of finite dimensional approximations of Kohn–Sham models. Adv. Comput. Math. 38 (2013) 225–256. | DOI | MR | Zbl

H. Chen and R. Schneider, Error estimates of some numerical atomic orbitals in molecular simulations. Commun. Comput. Phys. 18 (2015) 125–146. | DOI | MR | Zbl

H. Chen and R. Schneider, Numerical analysis of augmented plane wave methods for full-potential electronic structure calculations. ESAIM: M2AN 49 (2015) 755–785. | DOI | Numdam | MR | Zbl

G. Dusson and Y. Maday, A posteriori analysis of a nonlinear Gross–Pitaevskii-type eigenvalue problem. IMA J. Numer. Anal. 37 (2017) 94–137. | DOI | MR | Zbl

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. De Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R.M. Wentzcovitch, QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Cond. Mat. 21 (2009) 395502. | DOI

S. Goedecker, M. Teter and J. Hutter, Separable dual-space Gaussian pseudopotentials. Phys. Rev. B 54 (1996) 1703–1710. | DOI

X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Côté, T. Deutsch, L. Genovese, Ph. Ghosez, M. Giantomassi, S. Goedecker, D.R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M.J.T. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.-M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, M.J. Verstraete, G. Zerah and J.W. Zwanziger, ABINIT: First-principles approach to material and nanosystem properties. Comput. Phys. Commun. 180 (2009) 2582–2615. | DOI

X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-Y. Raty and D.C. Allan, First-principles computation of material properties: the ABINIT software project. Comput. Materials Sci. 25 (2002) 478–492. | DOI

F. Gygi and G. Galli, Real-space adaptive-coordinate electronic-structure calculations. Phys. Rev. B 52 (1995) R2229–R2232. | DOI

M. Hanrath, Wavefunction quality and error estimation of single- and multi-reference coupled-cluster and CI methods: the H4 model system. Chem. Phys. Lett. 466 (2008) 240–246. | DOI

T. Helgaker, P. Jørgensen and J. Olsen, Molecular electronic-structure theory. John Wiley & Sons, Ltd, Chichester, UK (2000).

J. Kaye, L. Lin and C. Yang, A posteriori error estimator for adaptive local basis functions to solve Kohn–Sham density functional theory. Commun. Math. Sci. 13 (2015) 1741–1773. | DOI | MR | Zbl

W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140 (1965) A1133–A1138. | DOI | MR

G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 5 (1996) 11169–11186. | DOI

W. Kutzelnigg, Error analysis and improvements of coupled-cluster theory. Theor. Chim. Acta 80 (1991) 349–386. | DOI

W. Kutzelnigg, Rate of convergence of basis expansions in quantum chemistry. AIP Conf. Proc. 1504 (2012) 15–30. | DOI

S. Li, K. Chen, M.-Y. Hsieh, N. Muralimanohar, C.D. Kersey, J.B. Brockman, A.F. Rodrigues and N.P. Jouppi, System implications of memory reliability in exascale computing. In Proc. of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis on - SC ’11, page 1, New York, New York, USA (2011). ACM Press.

L. Lin and B. Stamm, A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions Part I: Second order linear PDE. ESAIM: M2AN 50 (2016) 1193–1222. | DOI | Numdam | MR | Zbl

Y. Maday and G. Turinici, Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations. Numer. Math. 94 (2003) 739–770. | DOI | MR | Zbl

S. Mohr, L.E. Ratcliff, P. Boulanger, L. Genovese, D. Caliste, T. Deutsch and S. Goedecker, Daubechies wavelets for linear scaling density functional theory. J. Chemical Phys. 140 (2014) 204–110. | DOI

P. Motamarri, M.R. Nowak, K. Leiter, J. Knap and V. Gavini, Higher-order adaptive finite-element methods for Kohn–Sham density functional theory. J. Comput. Phys. 253 (2013) 308–343. | DOI | MR | Zbl

J.E. Pask and P.A. Sterne, Finite element methods in ab initio electronic structure calculations. Model. Simul. Mat. Sci. Eng. 13 (2005) R71–R96. | DOI

P. Pernot, B. Civalleri, D. Presti and A. Savin, Prediction uncertainty of density functional approximations for properties of crystals with cubic symmetry. J. Phys. Chemistry A 119 (2015) 5288–5304. | DOI

S.N. Pieniazek, F.R. Clemente and K.N. Houk, Sources of error in DFT computations of C–C bond formation thermochemistries: $\pi \to \sigma$ transformations and error cancellation by DFT methods. Angew. Chem. Int. Ed. 47 (2008) 7746–7749. | DOI

M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis. Vol. 53. Academic Press Inc., New York (1972). | MR | Zbl

T. Rohwedder and R. Schneider, Error estimates for the coupled cluster method. ESAIM: M2AN 47 (2013) 1553–1582. | DOI | Numdam | MR | Zbl

Y. Saad, J.R. Chelikowsky and S.M. Shontz, Numerical methods for electronic structure calculations of materials. SIAM Rev. 52 (2010) 3–54. | DOI | MR | Zbl